# Scaling limit of the uniform prudent walk

## TL;DR

This paper proves that the 2D uniform prudent walk is ballistic, predominantly follows one of four diagonals, and exhibits Gaussian fluctuations around the diagonal, revealing its large-scale behavior.

## Contribution

It establishes the ballistic nature and diagonal localization of the uniform prudent walk, along with a functional CLT for its fluctuations, advancing understanding of its scaling limit.

## Key findings

- Walk is ballistic and follows diagonals with equal probability
- Path fluctuations around the diagonal are Gaussian
- Provides a functional central limit theorem for the walk

## Abstract

We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [Bousquet-M\'elou, 2010], while another variant, the kinetic prudent walk has been analyzed in detail in [Beffara, Friedli and Velenik, 2010]. In this paper, we prove that the $2$-dimensional uniform prudent walk is ballistic and follows one of the $4$ diagonals with equal probability. We also establish a functional central limit theorem for the fluctuations of the path around the diagonal.